Internal Angles Article

[This article is the updated web version of an article that has
appeared in "Compass and Tape" in the US, and in
"Compass Points" in England.]

There seems to be a lot of confusion in the Cave Surveying community
about what to do with foresights and backsights when there are
magnetic anomolies. And I've even seen reports of folk saying that
nothing can be done with this situation and fore and back sights
don't help. This later view is often accompanied by an alleged proof
that shows that averaging foresights and backsights (or some other
mathematically meaningless or otherwise irrelevant method) is the wrong
thing to do. [In logic this fallacy is called a "straw man" argument,
where you make up something silly and claim it is implied by a view,
descredit the sillyness, and pretend you've discredited the original
view.]

So, here is explaination of how magnetic anomolies can be handled using
foresights and backsights, and how to do the error analysis when magnetic
anomolies are present. This will be shown on an example survey.

There can be [initially unknown] magnetic anomalies at every single
point. (Not an uncommon state of affairs for most lava tube surveys.)
No assumption is made as to the reason that these anomolies are present.
We do assume that magnetic north is in SOME direction at each point. (I.E.
the field strength is not zero.)

Note that because there are severe magnetic anomalies at all
points the foresights and backsights are going to be dramaticly
different. This is not wrong... they differ because magnetic
north at the various stations is different.

A survey instructor has pointed out that this data looks
much different than it would if it had came from a real surveyor.
In the interest of pushing cave surveyors back toward main line
surveying, I'll include a copy of the data as surveyors would
expect it to appear.

In order to do the processing we have to rearrange the data
from a shot specific viewpoint (as recorded) to a station
specific viewpoint, since the sights are actually relative
to magnetic north at the point taken, and not a
theoretical magnetic north for the shot itself.

One advantage of this is that it groups shots with
the same magnetic deviation.

All sightings taken from
exactly the same point share exactly the same
magnetic north, regardless of what direction that magnetic
north may be at that point.

Because of this we can now compute the change in angle (the "turned"
angle) between any two shots associated with a given point. This
angle is obviously independent of what direction magnetic north
points, as long as it points in some direction.

The turned angles, and an inital "basis of bearing" shot (or even
an assumed bearing) can be used to lay out the entire survey. And, if
there are loops in the data they can be used to do blunder analysis
of the loops, as shown below.

Since we are only concerned with computing internal angles
in this example, I'll drop the Distance and Inclination information.

From To Azimuth Comments
A B 214 (Foresight of shot from A to B)
A D 154 (Backsight of shot from D to A)
B A 42 (Backsight of shot from A to B)
B C 162 (Foresight of shot from B to C)
B D 102 (Backsight of shot from D to B)
C B 336 (Backsight of shot from B to C)
C D 36 (Foresight of shot from C to D)
D A 327 (Foresight of shot from D to A)
D B 267 (Foresight of shot from D to B)
D C 209 (Backsight of shot from D to C)

The pattern here is: list foresight if the shot was FROM this point,
list backsight if the shot was TO this point. We are
basically collecting all the measurements made at each
specific point, and making a record to go with that
specific point. (While magnetic north may be different
on both ends of a shot, in the absence of bad survey
procedure it should be the same for all the shots that were
taken from a single point.

Note that we have to be careful here about the order of shots,
angles, and labels. The angle "B-A-D" is *not* the same as the angle
"D-A-B", as one is the negative of the other. The label BAD, for example,
refers to the angle between B and D as measured from A (which is 214-154
= +60) and DAB is the angle between D and B (In that order) as measured
from A (Which is 154-214 = -60).

The internal angles sum to something other then a
multiple of 180. This loop contains at least
2 degrees of angle problems. It could be a single blunder of
two degrees, or it could have been (for example) a +8 degree
blunder and a -6 degree mistake canceling to be a two degree
problem.

In this specific case, we know that CDB is blundered, but only
because of the way this data was contrived. HOWEVER, in general
all we know is that there is a problem (or problems) somewhere
in this loop.

[Since the paragraph above was written, I've been taken to task
because there IS enough information to correctly identify the
blundered angle. For example, the book
[Wolf and Ghilani, 1997]
"Adjustment Computations, Statistics and Least Squares in Surveying and GIS",
in the chapter on blunder detection in horizontal surveys, gives a
technique for identifying the direction from points in the loop to
the specific station at fault. There are also tests documented in
the survey literature that give distance from stations in the loop
to the location of the blundered station.

We now know that the first loop in the example has no angle problems,
and that the second loop does.

Since the first loop has no angle problems, the discrepancy between
foresights and backsights in that loop must reflect real underlying
differences between magnetic north between the stations.

Simple arithmetic can now show that the difference in magnetic
north between A and B is 8 degrees (The discrepancy between fore and
back sights in shot AB), between A and D is -7 (The negative of the
discrepancy on shot from D to A) and of course between D and B it is
consistently 15 (The discrepancy from shot BD). (The difference
between magnetic norths is the negative of the shot discrepancy if
you are tracing the graph in a direction opposite the original shot.)

If one sets A as the reference, then it is easy to list
the differences for the whole of the (unblundered) net.
But magnetic north at C can only be estimated, since some shot to
C contains a blunder.

A major obvious assumption being made with that technique is
that all shots *FROM THE SAME POINT* have the same offset
from magnetic north. This is generally true unless the anomaly
is being caused by something the caver is carrying.

Clearly, if you went around averaging fore and back sights,
no reasonable analysis is possible. For shot AB this
would give you a recorded number of:

(azimuth_A + anomaly_A + azimuth_B + anomaly_B) / 2

which hopelessly intermingles the shots and any magnetic
anomalies.

Without loops you have no redundant information to check against,
so detecting blunders this way is not possible. (Although, obviously,
you can still correct for anomolies if there are no blunders)
HOWEVER, in surveys with few magnetic problems, there is sometimes some
blunder information to be gained. If one computes the (apparent) magnetic
anomalies in a traverse, the assumed magnetic north will be stable
but different on the two traverse pieces on each side of the blunder.
In most limestone areas this may aid in locating the blunder.
I really need to give another example to show this technique.